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%{\LARGE\bf 上海立信会计金融学院期终考试卷 --- 试题纸} \hspace{0.3cm} {\Large \underline{ A }卷 }
{\Large\bf \H 上海立信会计金融学院期终考试卷 } \hspace{0.3cm} {\Large \underline{ A }卷 }

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{\large \bf \H 2023 $\sim$ 2024 学年 第 二 学期 }

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{\large \bf \H \underline{ \emph{2021级数学与应用数学专业} } 《\underline{ \emph{多元统计分析} }》 课程代码：\underline{ 160290220 }  }

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{\H（本场考试属\underline{  开  }卷考试，考试时间共\underline{  90  }分钟，不准使用计算器）共\underline{  4  }页 }
%{\large （本场考试属闭卷考试，考试时间 90 分钟，禁止使用计算器） }

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%{\large 本考试卷共 4 页，请在本考试卷上答题。}

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班级 \underline{\hspace{3.5cm}} 学号 \underline{\hspace{3.5cm}} 姓名 \underline{\hspace{3.5cm}} 

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{\H
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%\caption{Nonlinear Model Results} % title of Table
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题号 &一&二&三&四&五&六&七&八&九&十&总分&合成人签名&审核人签名 \\
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%应得分&15&15&15&15&15&15&10&100 \\
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得分 $\,\,\,\,\,\,\,\,$ &&&&&&&&&&&&& \\
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本次考试共10题，每题10分。
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\begin{enumerate}

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\item %1 第2.2节：随机向量的数字特征：第33页例子2.2.3
设随机向量 $\vec{x}=(x_1,x_2)^t$ 的数学期望和协方差矩阵分别是
\begin{eqnarray*}
\vec{\mu}=\begin{pmatrix}2 \\ 3 \end{pmatrix}, \,\,\, 
\Sigma=\begin{pmatrix} 9&3 \\ 3&4 \end{pmatrix}.
\end{eqnarray*}
\begin{enumerate}%[label={(\arabic*)}]
\item  求 $\vec{x}$ 的相关矩阵 $R$. 
\item  设 $y_1=x_1-x_2, y_2=x_1+2x_2$, 求 $\vec{y} = (y_1,y_2)^t$ 的数学期望和协方差矩阵。
\end{enumerate}


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\item %2 第3.2节：多元正态分布的性质：第56页例子3.2.8
设 $\vec{x} = (x_1,x_2,x_3)^t\sim N_3(\mu,\Sigma)$, 这里 
$$
\vec{\mu}=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \,\,
\Sigma=\begin{pmatrix} 4 & 2 & 3 \\ 2 & 9 & 5 \\ 3 & 5 & 16 \end{pmatrix}. 
$$
求已知 $x_3$ 时，$(x_1, x_2)^t$ 的条件分布。


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\item %3 第3.4节：复相关系数：第61页复相关系数的定义
设有随机变量 $y$ 和随机向量 $\vec{x}=(x_1,\cdots,x_p)^t$, 设 
$$
E\begin{pmatrix} y \\ \vec{x} \end{pmatrix} = \begin{pmatrix} \mu_y \\ \mu_{\vec{x}} \end{pmatrix}, \,\,
V\begin{pmatrix} y \\ \vec{x} \end{pmatrix} = \begin{pmatrix} \sigma_{yy} & \sigma_{\vec{x}y}^t \\ \sigma_{\vec{x}y} & \Sigma_{\vec{x}\vec{x}}  \end{pmatrix}. 
$$
\begin{enumerate}
\item  求 $y$ 和线性组合 $\vec{\ell}\,^t \vec{x}$ 的相关系数的平方，其中 $\vec{\ell} = (\ell_1,\cdots,\ell_p)^t$ 是 $p$ 维常数向量。
\item  当 $\vec{\ell}$ 取何值时，相关系数的平方取最大值？
\end{enumerate} 


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\item %4 第4.2节：第87页单个总体均值向量的检验
设 $\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n$ 是取自多元正态总体 $\vec{x}\sim N_p(\mu,\Sigma)$ 的一个样本，设 $\Sigma>0$. 
分别在下述两种情况下，进行假设检验
$H_0: \vec{\mu}=\vec{\mu}_0, \,\,\, \mathrm{vs.} \,\,\, \vec{\mu}\neq \vec{\mu}_0. $
写出检验统计量和拒绝域。
\begin{enumerate}
\item  设 $\Sigma$ 已知。
\item  设 $\Sigma$ 未知。
\end{enumerate}


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\item %5 第4.3节：第95页两个总体均值的比较推断
设两个独立样本 $\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_{n_1}$ 和 $\vec{y}_1,\vec{y}_2,\cdots,\vec{y}_{n_2}$ 分别来自多元正态总体 $N_p(\vec{\mu}_1,\Sigma)$ 和 $N_p(\vec{\mu}_2,\Sigma)$. 考虑假设检验 $$H_0:\vec{\mu}_1=\vec{\mu}_2, \,\,\mathrm{vs.} \,\, H_1:\vec{\mu}_1\neq \vec{\mu}_2. $$
\begin{enumerate}
\item  写出检验统计量和拒绝规则。
\item  写出一切线性组合 $\{a^t(\vec{\mu}_1-\vec{\mu}_2),a\in\mathbb{R}^p\}$ 的置信度为 $1-\alpha$ 的联合置信区间。
\item  写出线性组合 $\{a_i^t(\vec{\mu}_1-\vec{\mu}_2),i=1,2,\cdots,k\}$ 的置信度为 $1-\alpha$ 的 Bonferroni 联合置信区间。
\end{enumerate}


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\item %6 第4.5节：第105页多个总体均值的比较检验
设有 $k$ 个总体 $\pi_1,\pi_2,\cdots,\pi_k$, 它们的分布分别是 $N_p(\vec{\mu}_1,\Sigma), N_p(\vec{\mu}_2,\Sigma), \cdots, N_p(\vec{\mu}_k,\Sigma)$. 从这 $k$ 个总体中各自独立地抽取一个样本，取自总体 $\pi_i$ 的样本为 $\vec{x}_{i1}, \vec{x}_{i2},\cdots,\vec{x}_{in_i}$, 这个样本的容量为 $n_i$.（这里的每个 $\vec{x}_{ij}$ 都是有 $p$ 个分量的列向量。）
考虑假设检验 $$H_0: \vec{\mu}_1=\vec{\mu}_2=\cdots=\vec{\mu}_k, \,\,\, \mathrm{vs.} \,\,\, H_1: \exists i\neq j, \,\mathrm{s.t.}\, \vec{\mu}_i\neq \vec{\mu}_j. $$
写出检验统计量和拒绝规则。



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\item %7 第5.2节：距离判别：第130页，两组距离判别
设两组 $\pi_1$ 和 $\pi_2$ 的均值分别为 $\vec{\mu}_1$ 和 $\vec{\mu}_2$, 协方差矩阵分别为 $\Sigma_1$ 和 $\Sigma_2$. 
设 $\Sigma_1=\Sigma_2=\Sigma$. 设 $\vec{x}$ 是一个新的样品，使用距离判别法判断它来自哪一组。
\begin{enumerate}
\item  写出平方马氏距离的计算公式和判别规则。
\item  设两组均为正态总体，写出误判概率 $P(2\mid 1)$ 和 $P(1\mid 2)$ 的计算公式。
\end{enumerate}



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\item %8 第6.3节：系统聚类法：最短距离法 
考虑使用最短距离法的系统聚类法。
\begin{enumerate}
\item  两类之间的距离是怎么定义的？
\item  这种聚类分析的步骤是什么？
\end{enumerate}



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\item %9 第7.2节：总体的主成分：第一主成分和第二主成分的计算方法
设 $\vec{x}=(x_1,x_2,\cdots,x_p)^t$ 是一个 $p$ 维随机向量，设均值向量为 $\vec{\mu} = E(\vec{x})$, 协方差矩阵为 $\Sigma=V(\vec{x})$. 
写出下述第一主成分和第二主成分的计算步骤，
\begin{equation*}
\left\{
\begin{aligned}
y_1 &= a_{11}x_1 + a_{21}x_2 + \cdots + a_{p1}x_p = \vec{a}_1^t \vec{x}, \\ 
y_2 &= a_{12}x_1 + a_{22}x_2 + \cdots + a_{p2}x_p = \vec{a}_2^t \vec{x}. 
\end{aligned}
\right.
\end{equation*}




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\item %10 第8.3节：因子模型的参数估计：第257页，主成分法
设有 $p$ 维可观测随机向量 $\vec{x}=(x_1,x_2,\cdots,x_p)^t$, 
设 $\vec{x}_1,\vec{x}_2,\cdots,\vec{x}_n$ 是一组 $p$ 维样本。
\begin{enumerate}
\item  写出均值向量 $\vec{\mu}$ 和协方差矩阵 $\Sigma$ 的估计。 
\item  写出正交因子模型 $\vec{x}=\vec{\mu}+A\vec{f}+\vec{\varepsilon}$ 的假设条件并简要解释。
\item  使用主成分法估计因子载荷矩阵 $A$ 与特殊方差矩阵 $D$. 
\end{enumerate}


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